Average Error: 22.2 → 0.2
Time: 15.7s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -198843285.268944234 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -198843285.268944234 \lor \neg \left(y \le 212488777.898407\right):\\
\;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\

\mathbf{else}:\\
\;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\

\end{array}
double f(double x, double y) {
        double r630767 = 1.0;
        double r630768 = x;
        double r630769 = r630767 - r630768;
        double r630770 = y;
        double r630771 = r630769 * r630770;
        double r630772 = r630770 + r630767;
        double r630773 = r630771 / r630772;
        double r630774 = r630767 - r630773;
        return r630774;
}
↓
double f(double x, double y) {
        double r630775 = y;
        double r630776 = -198843285.26894423;
        bool r630777 = r630775 <= r630776;
        double r630778 = 212488777.89840698;
        bool r630779 = r630775 <= r630778;
        double r630780 = !r630779;
        bool r630781 = r630777 || r630780;
        double r630782 = 1.0;
        double r630783 = r630782 / r630775;
        double r630784 = x;
        double r630785 = r630783 + r630784;
        double r630786 = r630784 / r630775;
        double r630787 = r630782 * r630786;
        double r630788 = r630785 - r630787;
        double r630789 = r630782 - r630784;
        double r630790 = r630789 * r630775;
        double r630791 = r630775 + r630782;
        double r630792 = r630790 / r630791;
        double r630793 = r630782 - r630792;
        double r630794 = r630781 ? r630788 : r630793;
        return r630794;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.2
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -198843285.26894423 or 212488777.89840698 < y

    1. Initial program 45.7

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

    2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(x + 1 \cdot \frac{1}{y}\right) - 1 \cdot \frac{x}{y}}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}}\]

    if -198843285.26894423 < y < 212488777.89840698

    1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -198843285.268944234 \lor \neg \left(y \le 212488777.898407\right):\\ \;\;\;\;\left(\frac{1}{y} + x\right) - 1 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))